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SC17-005: Implementation of the DPG Method in a FE Code Supporting H1, H(curl), H(div), and L2-Conforming Finite Elements

Leszek Demkowicz, The University of Texas at Austin
Stefan Henneking, The University of Texas at Austin
 

The four-lectures course is addressed to practitioners of standard Finite Element (FE) methods familiar with basic variational formulations, the (Bubnov–)Galerkin method and the standard technology of FEs. The class combines a short introduction to the “Discontinuous Petrov–Galerkin (DPG) Method with Optimal Test Functions” with a crash course on the energy spaces forming the exact sequence and the corresponding conforming FE discretizations. We will introduce the participants to hp3D—a 3D MPI/OpenMP code supporting hp-discretizations of the exact-sequence elements on hybrid (tets, cubes, prisms, pyramids) meshes and demonstrate how to implement the DPG method in such a framework. On the application side, we will focus on wave propagation problems: time-harmonic acoustics, Maxwell’s equations, and elastodynamics.

Syllabus:

  1. Lecture 1
    1. Examples of variational formulations with symmetric and non-symmetric functional setting; brief introduction to energy spaces.
    2. A crash course on H1, H(curl), H(div), and L2-conforming finite elements.
  2. Lecture 2
    1. Introduction to the hp3D code.
    2. Examples of applications of the Bubnov–Galerkin method.
  3. Lecture 3
    1. A crash course on the DPG method.
    2. Implementation of DPG in the hp3D code.
  4. Lecture 4
    1. MPI/OpenMP parallel computation with the hp3D code.
    2. Examples of applications of the DPG method.

Course Material:

  1. L. Demkowicz. Lecture Notes on Energy Spaces. Technical Report 13, ICES, 2018. https://www.oden.utexas.edu/media/reports/2018/1813.pdf
  2. L. Demkowicz. Lecture Notes on Mathematical Theory of Finite Elements. Technical Report 11, Oden Institute, June 2020. https://www.oden.utexas.edu/media/reports/2020/2011.pdf
  3. S. Henneking and L. Demkowicz. Computing with hp Finite Elements III. Parallel hp Code. 2022. In preparation, available upon request.

Software: